Algorithmic Number Theory in Function Fields

Algorithmic Number Theory in Function Fields Renate Scheidler UNCG Summer School in Computational Number Theory 2016: Function Fields May 30 June 3, 2016

References Henning Stichtenoth, Algebraic Function Fields and Codes, second ed., GTM vol. 54, Springer 2009 Michael Rosen, Number Theory in Function Fields, GTM vol. 210, Springer 2002 Gabriel Daniel Villa Salvador, Topics in the Theory of Algebraic Function Fields, Birkhäuser 2006 Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 2 / 92

Valuation Theory

Absolute Values Throughout, let F be a field. Definition An absolute value on F is a map F R such that for all a, b F : a 0, with equality if and only if a = 0 ab = a b a + b a + b (archimedian) or a + b max{ a, b } (non-archimedian) Examples The well-known absolute value on Q (or on R or on C) is an archimedian absolute value in the sense of the above definition. The trivial absolute value on any field F, defined via a = 0 when a = 0 and a = 1 otherwise, is a non-archimedian absolute value. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 4 / 92

p-adic Absolute Values on Q Let p be ay prime number, and define a map p on Q as follows: For r Q, write r = p n a b with n Z and p ab and set r p = p n. Then p is a non-archimedian absolute value on Q, called the p-adic absolute value on Q. Theorem (Ostrowski) The p-adic absolute values, along with the trivial and the ordinary absolute value, are the only valuations on Q. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 5 / 92

Rational Function Fields Notation For any field K: K[x] denotes the ring of polynomials in x with coefficients in K. K(x) denotes the field of rational functions in x with coefficients in K: K(x) = { f (x) f (x), g(x) K[x] with g(x) 0}. g(x) Note that F = K(x) is our first example of an algebraic function field. More formally: Definition A rational function field F /K is a field F of the form F = K(x) where x F is transcendental over K. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 6 / 92

Absolute Values on K(x) Fix a constant c R, c > 1, and let r(x) K(x) be nonzero. p-adic absolute values on K(x): Let p(x) be any monic irreducible polynomial in K[x], and write r(x) = p(x) n a(x)/b(x) with n Z and p(x) a(x)b(x). Define r(x) p(x) = c n. Then p(x) is a non-archimedian absolute value on K(x). Infinite absolute value on K(x): Write r(x) = f (x)/g(x) and define r(x) = c deg(f ) deg(g). Then is a non-archimedian absolute value on K(x). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 7 / 92

Remarks on Absolute Values on K(x) These, plus the trivial absolute value, are essentially all the absolute values on K(x), up to trivial modifications such as using a different constant c, using a different normalization on the irreducible polynomials p(x). All absolute values on K(x) are non-archimedian (different from Q!) When K = F q is a finite field of order q, one usually chooses c = q. When K is a field of characteristic 0, one usually chooses c = e = 2.71828.... Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 8 / 92

Valuations Definition A valuation on F is a map v F R { } such that for all a, b F : v(a) = if and only if a = 0 v(ab) = v(a) + v(b) v(a + b) min{v(a), v(b)} The pair (F, v) is called a valued field. (Here, n and + = + n = for all n Z.) Remark Let c > 1 be any constant. Then v is a valuation on F if and only if = c v( ) is a non-archimedian absolute value on F (with c = 0). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 9 / 92

Examples Trivial valuation: for any a F, define v(a) = when a = 0 and v(a) = 0 otherwise. Then v is a valuation on F. p-adic valuations on Q: for any prime p and r = p n a/b Q, define v p (r) = n. Then v p is a valuation on Q. p-adic valuations on K(x): for any monic irreducible polynomial p(x) K[x] and r(x) = p(x) n a(x)/b(x) K(x), define v p(x) (r(x)) = n. Then v p(x) is a valuation on K(x). Infinite valuation on K(x): for r(x) = f (x)/g(x) K(x), define v (r(x)) = deg(g) deg(f ). Then v is a valuation on K(x). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 10 / 92

More on Valuations Definition A valuation v is discrete if it takes on values in Z { } and normalized if there exists an element u F with v(u) = 1. Such an element u is a uniformizer (or prime element) for v. Remarks All four valuations from the previous slide are discrete. Every p-adic valuation on Q is normalized with uniformizer p. Every p-adic valuation on K(x) is normalized with uniformizer p(x). The infinite valuation on K(x) is normalized with uniformizer 1/x. The p-adic and infinite valuations on K(x) all satisfy v(a) = 0 for all a K. They constitute all the valuations on K(x) with that property. Remark A discrete valuation is normalized if and only if it is surjective. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 11 / 92

Valuation Rings For a discretely valued field (F, v), define the following subsets of F : O v = {a F v(a) 0}, O v = {a F v(a) = 0}, P v = {a F v(a) > 0} = O v O v. F v = O v /P v. Properties: O v is an integral domain and a discrete valuation ring, i.e. O v F and for a F, we have a O v or a 1 O v. O v is the unit group of O v. P v is the unique maximal ideal of O v ; in particular, F v is a field called the residue field of v. Every a F has a unique representation a = ɛu n with ɛ O v and n = v(a) Z. O v is principal ideal domain whose ideals are generated by the non-negative powers of u; in particular, u is a generator of P v. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 12 / 92

Example: p-adic Valuations For any p-adic valuation v p on Q: O vp = {r Q r = a/b with gcd(a, b) = 1 and p b} Ov p = {r Q r = a/b with gcd(a, b) = 1 and p ab} P vp = {r Q r = a/b with gcd(a, b) = 1, p a, p b} F vp = F p. Similarly, for any p-adic valuation v p(x) on K(x): O vp(x) = {r(x) K(x) r(x) = a(x)/b(x) with gcd(a, b) = 1, p(x) b(x)} Ov p(x) = {r(x) K(x) r(x) = a(x)/b(x) with gcd(a, b) = 1, p(x) a(x)b(x)} P vp(x) = {r(x) K(x) (x) = a(x)/b(x) with gcd(a, b) = 1, p(x) a(x), p(x) b(x)} F vp(x) = K[x]/(p(x)) where (p(x)) is the K[x]-ideal generated by p(x) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 13 / 92

Example: Infinite Valuation on K(x) For the infinite valuation v on K(x): O v = {r(x) K(x) r(x) = f (x)/g(x) with deg(f ) deg(g)} Ov = {r(x) K(x) r(x) = f (x)/g(x) with deg(f ) = deg(g)} P v = {r(x) K(x) (x) = f (x)/g(x) with deg(f ) < deg(g)} F v = K We will henceforth write O, P, F for brevity. Example v ( x 7 2x 3 + 3x ) = 2 and x 7 2x 3 + 3x = ( 1 x ) 2 x 3 7x 2 2x 3. + 3 O Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 14 / 92

Places Definition A place of F is the unique maximal ideal of a discrete valuation ring in F. The set of places of F is denoted P(F ). Theorem There is a one-to-one correspondence between the set of normalized discrete valuations on F and the set P(F ) of places of F as follows: If v is a normalized discrete valuation on F, then P v P(F ) is the unique maximal ideal in the discrete valuation ring O v. If P is a place of F, then the discrete valuation ring O F containing P as its unique maximal ideal is determined, and P defines a discrete normalized valuation on F as follows: if u is any generator of P, then every element a F has a unique representation a = ɛu n with n Z and ɛ a unit in O, and we define v(a) = n and v(0) =. Note that u is a uniformizer for v. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 15 / 92

Examples of Places For any prime number p, the set P = {r Q r = a/b with gcd(a, b) = 1, p a, p b} = P vp is a place of Q with corresponding valuation v p. The set P(K(x)) consists of the finite places of K(x) of the form P p(x) = P vp(x) where p(x) is a monic irreducible polynomial in K[x] and the infinite place of K(x) of the form P = P v. Let F /Q be a number field with ring of integers O F (the integral closure of Z in F ). Then every prime ideal in O F is a place of F. Let F be a finite algebraic extension of F q (x) and let O F be the integral closure of the polynomial ring F q [x] in F. Then every prime ideal in O F is a place of K. Note that there are other places of F that do not arise in this way (more on this later). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 16 / 92

Function Fields

Function Fields Definition Let K be a field. An algebraic function field F /K in one variable over K is a field extension F K such that F is finite algebraic extension of K(x) for some x F that is transcendental over K. We will shorten this terminology to just function field. In other words, a function field is of the form F = K(x, y) where x F is transcendental over K, y F is algebraic over K(x), so there exists a monic irreducible polynomial φ(y ) K(x)[Y ] of degree n = [F K(x)] with φ(y) = 0. Remark It is important to note that there are many choices for x, and the degree [F K(x)] may change with the choice of x. This is different from number fields where the degree over Q is fixed. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 18 / 92

Examples of Function Fields A function field is rational if F = K(x) for some element x F that is transcendental over K. The meromorphic functions on a compact Riemann surface form a function field over C (the complex numbers). Let E y 2 = x 3 + Ax + B be an elliptic curve defined over a field K of characteristic different from 2 and 3. Then F = K(x, y) is a function field over K. Note that [F K(x)] = 2 and [F K(y)] = 3. More generally, consider the curve y 2 = f (x) where f (x) K[x] is a square-free polynomial and K has characteristic different from 2. Then F = K(x, y) is a function field over K whose elements are of the form F = { a(x) + b(x)y a(x), b(x) K(x) }. Note that [F K(x)] = 2 and [F K(y)] = deg(f ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 19 / 92

Function Fields of Curves Definition A plane affine irreducible algebraic curve over a field K is the zero locus of an irreducible polynomial Φ(x, Y ) in two variables with coefficients in K. We will shorten this terminology to just curve. Definition The coordinate ring of a curve C Φ(x, y) = 0 over a field K is the ring K[x, Y ]/(Φ(x, Y )) where (Φ(x, y)) is the principal K[x, y]-ideal generated by Φ(x, y). The function field of C is the field of fractions of its coordinate ring. Remark: The function field of a curve is a function field as defined previously. Conversely, every function field F /K is the function field of the curve given by a minimal polynomial of F /K(x). Note that a function field has many defining curves. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 20 / 92

Constant Fields Definition The constant field of a function field F /K is the algebraic closure of K in F, i.e. the field K = {z F z is algebraic over K}. F /K is a geometric function field if K = K. Sometimes K is called the full or the exact field of constants of F /K. Remark K K F, and every element in F K is transcendental over K. Remark Write F = K(x, y). Then F /K is a geometric function field if and only if the minimal polynomial of y over K(x) is absolutely irreducible, i.e. irreducible over K(x) where K is the algebraic closure of K. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 21 / 92

Examples K(x)/K is always geometric. If K is algebraically closed (e.g. K = C), then any F /K is geometric. Let F = K(x, y) where y 2 = f (x) with f (x) K[x] square-free. Then F /K(x) is geometric if and only if f (x) is non-constant. Suppose 1 is not a square in K (e.g. K = R or K = F q with q 3 (mod 4)), and let F = K(x, y) where x 2 + y 4 = 0. Then K = K(i) where i K is a square root of 1. So [ K K] = 2, and F /K is not geometric. Note that [F K(x)] = 4 and [F K(x)] = [ K(x) K(x)] = 2. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 22 / 92

Residue Fields and Degrees Recall that a place P of a field F is the unique maximal ideal of some discrete valuation ring O P of F, and its residue field is F P = O P /P. Remark: K O P for all P P(F ). Definition Let F /K be a geometric function field and P a place of F. Then the degree of P is the field extension degree deg(p) = [F P K]. Places of degree one are called rational. The set of rational places of F is denoted P 1 (F ). Remark deg(p) [F K(x)] for any x P, so deg(p) is always finite. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 23 / 92

Example: Residue Fields of Places of K(x) For any finite place P p(x) of K(x), a K basis of F P is {1, x,..., x deg(p) 1 }, so deg(p p(x) ) = deg(p). For the infinite place P of K(x), we have F P = K and hence deg(p ) = 1. K is algebraically closed if and only if the finite places of K(x) correspond exactly the linear polynomials x + α with α K, i.e. if and only if all the places of K(x) are rational, so P(K(x)) = P 1 (K(x)). In this case, there is a one-to-one correspondence between P 1 (K(x)) and the points on the projective line P 1 (K) = K { } via P 1 (K(x)) P 1 (K(x)) via x + α α, 1/x. Hence the name infinite place think of this as substituting x = 0 into the uniformizer. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 24 / 92

Values of Functions Recall that h(x) = q(x)(x α) + h(α) for all h(x) K[x] and α K. One can think of h(a) = h(x)(mod x α) as the value of h(x) at the place x α. For h(x) K, this value is the same for any α, i.e. constant. More generally, for any place P of a function field F /K, one can interpret cosets z + P as values z(p) which are constant for elements z K. Hence the terms function field and constant field. Let r(x) = f (x)/g(x) K(x) with gcd(f, g) = 1. Consider a finite place P x α (α K). Then v x α (r(x)) > 0 f (α) = 0, g(α) 0 α is a zero of r(x). v x α (r(x)) < 0 g(α) = 0, f (α) 0 α is a pole of r(x). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 25 / 92

Zeros and Poles Definition Let P be a place of a function field F /K and z F. Then P is a zero of z if v P (z) > 0 and a pole of z if v P (z) < 0. More generally, for any positive integer m, P is a zero of z of order (or multiplicity) m if v p (z) = m and a pole of z of order m if v P (z) = m. Properties: Every transcendental element in F /K has at least one zero and at least one pole. In particular, if F /K is geometric, then every element in F K has at least one zero and at least one pole. Every non-zero element in F has only finitely many zeros and poles. When counted with multiplicities, every non-zero element in F has the same number of zeros and poles. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 26 / 92

Divisors and Genus

Recollection: Ideals in Number Fields Recall that in a number field: Every ideal in the ring of integers has a unique factorization into prime ideals. By allowing negative exponents, this extends to fractional ideals. So the prime ideals generate the group of fractional ideals. Two non-zero fractional ideals are equivalent if they differ by a factor that is a principal ideal. The ideal class group is the group of non-zero fractional ideals modulo (principal) equivalence whose order is class number of the field. It is a finite abelian group that is an important invariant of the field. We now consider analogous notions in function fields, with prime ideals replaced by places, and multiplication (products) replaced by addition (sums). Assume henceforth that F /K is a geometric function field. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 28 / 92

Divisors Definition The Divisor group of F /K, denoted Div(F ), is the free group generated by the places of F /K. Its elements, called divisors of F, are formal finite sums of places. Let Then D = n P P with n P Z and n P = 0 for almost all P P(F ). P P(F ) the value of D at P is v P (D) = n P for any P P(F ). the support of D is supp(d) = {P P(F ) v P (D) 0}. the degree of D is deg(d) = P P(F ) n P deg(p). D is a prime divisor if it is of the form D = P for some P P(F ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 29 / 92

More on Divisors Remarks Every divisor is a unique sum of finitely many prime divisors (note that some prime divisors in the support may have negative coefficients). The notions of value and degree are compatible with their previous definitions. In particular: For any place P of F, the normalized discrete valuation on F associated to P extends to a group homomorphism v P Div(F ) Z { }. The degree map defined on places of F extends to a group homomorphism deg Div(F ) Z { } whose kernel is the subgroup Div 0 (F ) of Div(F ) consisting of all degree zero divisors. F. K. Schmidt proved that every function field F over a finite field K = F q has a divisor of degree one, so in this case, the degree homomorphism on Div(F ) is surjective. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 30 / 92

Principal Divisors Definition A divisor D Div(F ) is principal if it is of the form D = for some z F. Write D = div(z). Definition v P (z)p P P(F ) The zero divisor and pole divisor of a principal divisor div(z) are the respective divisors div(z) 0 = So div(z) = div(z) 0 div(z). v P (z)p, div(z) = v P (z)p. v P (z)>0 v P (z)<0 Example: In F = K(x), we have div(x) 0 = P x and div(x) = P. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 31 / 92

More on Principal Divisors Theorem Let x F K. Then deg(div(x) 0 ) = deg(div(x) ) = [F K(x)]. It follows that deg(div(z)) = 0, so the principal divisors form a subgroup of Div 0 (F ), denoted Prin(F ). Definition Two divisors D 1, D 2 Div(F ) are (linearly) equivalent, denoted D 1 D 2, if D 1 D 2 Prin(F ). Remark and Notation Linear equivalence is an equivalence relation. The class of a divisor D under linear equivalence is denoted [D]. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 32 / 92

Class Group and Zero Class Group Definition The factor groups Cl(F ) = Div(F )/ Prin(F ) and Cl 0 (F ) = Div 0 (F )/ Prin(F ) are the divisor class group and the degree zero divisor class group of F /K, respectively. (Usually abbreviated to just class group and zero class group.) Remarks and Definition Both Cl(F ) and Cl 0 (F ) are abelian groups. Cl(F ) is always infinite, but Cl 0 (F ) may or may not be infinite. It it is finite, then the order h F is called the degree zero divisor class number (or just class number) of F /K. We will see later on that h F is always finite for a function field over a finite field. Sometimes the zero class group is also called the Jacobian of F /K. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 33 / 92

More on Class Groups Theorem Every rational function field has class number one. Remark There are 8 non-rational function fields F /F q of class number one. All have q 4, and defining curves for all of them are known. Theorem Let F /K be a non-rational function field that has a rational place, denoted P. Then the map Φ P 1 (F ) Cl 0 (F ) via P [P P ] is injective. The above embedding imposes an abelian group structure on P 1 (F ). Note that this group structure is non-canonical (depends on the choice of P ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 34 / 92

Effective Divisors Definition Define a partial order on Div(F ) via D 1 D 2 v P (D 1 ) v P (D 2 ) for all P P(F ). A divisor D Div(F ) is effective (or integral or positive) if D 0. Examples The trivial divisor D = 0 is effective. Every prime divisor is effective. The zero and pole divisors of a principal divisor are effective. The sum of two effective divisors is effective. So the effective divisors form a sub-monoid of Div(F ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 35 / 92

Riemann-Roch Spaces Definition The Riemann-Roch space of a divisor D Div(F ) is the set L(D) = {x F div(x) + D 0} {0}. Interpretation L(D) consists of all x F such that If D has a pole of order m at P, then x has a zero of order at least m at P. If D has a zero of order m at P, then x may have a pole at P, but its order cannot exceed m. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 36 / 92

Examples Examples Let F = K(x). L(nP ) = {f (x) K[x] deg(f ) n} for n 0. If D = 3P x 1 + 2P x 2 + 4P x 7, then (x 1) 3 L(D) = { (x 2) 2 r(x) r(x) K(x), deg(r) 3}. (x 7) 4 For any function field F /K, if P P(F ) and n Z with n > 0, then L(nP) L((n 1)P) = {x F div(x) = np}. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 37 / 92

Properties of Riemann-Roch Spaces x L(D) if and only if v P (x) v P (D) for all P P(F ). L(D) is a K-vector space. If D 1 D 2, then L(D 1 ) L(D 2 ) (isomorphic as K-vector spaces). L(0) = K. If deg(d) < 0 or D Div 0 (F ) Prin(F ), then L(D) = {0}. L(D) {0} if and only if the class [D] contains an effective divisor. Notation The K-vector space dimension of L(D) is denoted l(d) = dim K (L(D)). Remark Both deg(d) and l(d) depend only on the divisor class [D]. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 38 / 92

Genus Definition The genus of F /K is g = max{deg(d) l(d) + 1 D Div(F )}. Remark g 0. (Because deg(0) l(0) + 1 = 0.) Theorem (Hasse-Weil) Let F /F q be a function field of genus g over a finite field of order q. q + 1 2g q P 1 (F ) q + 1 + 2g q (so P 1 (F ) q for q large). ( q 1) 2g Cl 0 (F ) ( q + 1) 2g (so Cl 0 (F ) q g for q large). The class group and genus are important invariants of any function field! Unfortunately, they are not easy to compute... Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 39 / 92

Riemann-Roch Theorem Theorem (Riemann-Roch) There exist a divisor class W Cl(D) such that for all D Div(F ) and all W W, we have l(w D) = g 1 + l(d) deg(d). Remarks and Definition W is unique with this property. W is called the canonical divisor class and its elements are called canonical divisors. W can be explicitly described as the class of Weil differentials of F. Corollary deg(w ) = 2g 2 and l(w ) = g for any canonical divisor W. If deg(d) 2g 1, then l(d) = deg(d) g + 1. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 40 / 92

Function Field Extensions

Recollection: Prime Ideals in Number Fields Recall that in a number field extension F /F /Q: A prime ideal p of O F need not remain prime when extended to O F. Rather, it has a prime ideal factorization po F = P e 1 1 Pe 2 2 Per r in O F. Each P i is said to lie above p, written P i p. Finitely many prime ideals of O F lie above any prime ideal of O F. p is said to lie below each P i. A unique prime ideal of O F lies below every prime ideal of O F. e i is called the ramification index of P i p. The field extension degree f i = [O F /P O F /p] is called the residue degree of P i p. The norm of P i is the O F -ideal N F /F (P i ) = p f i. The norm extends multiplicatively to all ideals of O F. The fundamental identity r i=1 e i f i = [F F ] holds. Once again, we consider analogous notions in function field extensions, with prime ideals replaced by places, and products replaced by sums. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 42 / 92

Function Field Extensions Notation and Assumption F /K and F /K are geometric function fields with F F and K K. F is a finite algebraic extension of F. K is perfect, i.e. every irreducible polynomial in K[x] has distinct roots. Remark: Finite fields, algebraically closed fields, and characteristic 0 fields are all perfect. K = F q (x) is not perfect (consider φ(t ) = T q x). Theorem and Definition Every place P of F contains a unique place P of F, namely P = P F. Write P P. For every place P of F, P P for only finitely many places P of F. P P if and only if O P = O P F. In this case O P is an O P -module of rank [F F ]. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 43 / 92

Ramification, Residue Degree, Norm and Co-Norm Theorem and Definition Let P P(F ), P P(F ) with respective discrete normalized valuations v P, v P and residue fields F P = O P /P, F P = O P /P. Assume P P. There is a unique positive integer e = e(p P) such that v P (x) = ev P (x) for all x F, called the ramification index of P P. There is a natural embedding F P F P via x + P x + P. The extension degree f (P P) = [F P F P ] is called the residue degree (or relative degree) of P P. The norm of P is the divisor N F /F (P ) = f (P P)P of F. The co-norm of P is the divisor con F /F (P) = e(p P)P of F. P P The norm and co-norm extend additively to homomorphisms on divisors and respect principal divisors, so they also extend to homomorphisms on divisor classes. e(p P)f (P P) = [F F ] (fundamental identity). P P Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 44 / 92

Example Quadratic Extensions Let char(k) 2, F = K(x, y) where x F is transcendental over K and y 2 = f (x) with f (x) K[x] K 2 square-free. For any place P of F, with P = P K(x): 2v P (y) = v P (y 2 ) = v P (f ) = e(p P)v P (f ). Case P is finite. Write P = P p(x) with p(x) K[x] monic and irreducible. If p(x) f (x) then v P (y) = 0. If p(x) f (x), then v P (f ) = 1 (since f is square-free), so e(p P) = 2. Then v P (y) = 1 and Con F /K(x) (P) = 2P. Case P = P. Then v P (f ) = v (f ) = deg(f ). If deg(f ) is odd, then e(p P) = 2. Then v P (y) = deg(f ) and Con F /K(x) (P) = 2P. If deg(f ) is even, then we will later see that e(p P) = 1. Then v P (y) = deg(f )/2. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 45 / 92

Extension Terminology Definition Let P P(F ). If P P(F ) with P P, then P lies above P and P lies below P. P is unramified if e(p P) = 1 for all P P and ramified otherwise. P is wildly ramified if char(k) divides e(p P) for some P P, and tamely ramified otherwise. P is totally ramified if Con F /F (P) = [F F ]P. P is inert in F if Con F /F (P) = P with f (P P) = [F F ]. P splits completely in F if e(p P) = f (P P) = 1 for all P P. Sufficient (but not necessary) conditions for a function field to be tamely ramified are: char(k) = 0. [F F ] < char(k) when char(k) is positive.) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 46 / 92

Properties Properties relating Upstairs to Downstairs : deg(p ) = f (P P) [K K] deg(p) for all P P(F ), P P(F ) with P P. deg(con F /F (D)) = [F F ] [K deg(d) for all D Div(F ). K] N F /F (Con F /F (D)) = [F F ] D for all D Div(F ). Transitivity Theorem Consider F /K, F /K and F /K with F F F and K K K. Let P P(F ), P P(F ), P P(F ) with P P P. Then e(p P) = e(p P )e(p P) and f (P P) = f (P P )f (P P). N F /F (D ) = N F /F (N F /F (D )) for all D Div(F ). Con F /F (D) = Con F /F (Con F /F (D)) for all D Div(F ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 47 / 92

Recollection: Prime Splitting in Number Fields In number fields, prime decomposition can almost always be obtained from Kummer s Theorem (Number Fields) Let F /Q be a number field and p a prime. Let F = Q(α) such that α O F and p does not divide the index [O F Z[α]]. Let φ α (T ) Z[T ] be the minimal polynomial of α and consider φ α (T ) F p [T ], the reduction of φ α (T ) modulo p. Let φ α (T ) φ 1 (T ) e 1 φ 2 (T ) e 2 φ r (T ) er (mod p) be the factorization of φ α (T ) over F p into distinct irreducible polynomials. Then the prime ideal factorization of p in O F is po F = p e 1 1 pe 2 2 per r, where p i = po F + φ i (α)o F, and p i has residue degree f i = deg(φ i ), for 1 i r. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 48 / 92

Kummer s Theorem for Function Fields Kummer s Theorem (Function Fields) Let F /F be a finite algebraic extension of function fields of degree n = [F F ] and P P(F ). Let F = F (y) such that y is integral over O P. Let φ y (T ) O P [T ] be the minimal polynomial of y over F, and consider φ y (T ) F p [T ], the image of φ y (T ) modulo P under the residue map O P F P. Let φ y (T ) φ 1 (T ) ɛ 1 φ 2 (T ) ɛ 2 φ r (T ) ɛr (mod P) be the factorization of φ y (T ) over F P into distinct irreducible polynomials. Then there are at least r distinct places P 1, P 2,..., P r P(F ) lying above P. Furthermore, φ i (y) P i and f (P i P) deg(φ i) for 1 i r. If in addition, {1, y,..., y n 1 } is an O P -basis of the integral closure O P = r i=1 O P i in F or ɛ i = 1 for 1 i r, then Con F /F (P) = ɛ 1 P 1 + ɛ 2P 2 + + ɛ r P r, so e(p i P) = ɛ i and f (P i P) = deg(φ i) for 1 i r. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 49 / 92

Example Quadratic Extensions Let char(k) 2, F = K(x, y) where x F is transcendental over K and y 2 = f (x) with f (x) K[x] K 2 square-free. A finite place P = P p(x) of K(x) ramifies in F if p(x) divides f (x); splits in F if f (x) is a square modulo p(x); is inert in F if f (x) is a non-square modulo p(x). The infinite place of K(x) ramifies in F if deg(f ) is odd; splits in F if deg(f ) is even and sgn(f ) is a square in K ; is inert in F if deg(f ) is even and sgn(f ) is a non-square in K. E.g. if K = F 5 and f (x) = x 3 + 3x + 2 = (x + 1)(x + 2) F 5 [x], then the ramified places of F 5 (x) are P x+1, P x+2 and P. The place P x 2 +2x+4 of F 5 (x) splits in F because x 2 + 2x + 4 = (2x 2 + 2) 2 + x(x 3 + 3x + 2) (2x 2 + 2) 2 (mod f (x)). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 50 / 92

The Different Assume that F /F is a tamely ramified function field extension. Definition The different (or ramification divisor) of F /F is Example Diff(F /F ) = P P(F ) P P (e(p P) 1)P Div(F ). F = K(x, y) with y 2 = f (x) = p 1 (x) p r (x) (prime factorization of f (x)). Diff(F /K(x)) = P p 1 (x) + + P p r (x) + δp where P p i (x) is the unique place lying above P p i (x); P is the unique place lying above P when P is ramified; δ {0, 1} is the parity of deg(f ). It follows that deg(diff(f /K(x))) = deg(f ) + δ (an even integer). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 51 / 92

Genus and Ramification Theorem (Hurwitz Genus Formula) Let F /K be a finite algebraic function field extension of F /K, and denote by g F and g F the genera of F and F, respectively. Then 2g F 2 = [F F ] [K K] (2g F 2) + deg(diff(f /F )). Corollary The different has even degree. If x F K, then g F = 1 2 deg(diff(f /K(x)) [F K(x)] + 1. Example (elliptic and hyperelliptic function fields): F = K(x, y) with char(k) 2; y 2 = f (x) with f (x) K[x] square-free. g = (deg(f ) 1)/2 (so deg(f ) = 2g + 1 or 2g + 2). deg(diff(f /K(x)) = 2g + 2. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 52 / 92

Constant Field Extensions Recall our assumptions that K is perfect and K /K is algebraic. Definition An extension F /K of F /K is a constant field extension if F = FK. Theorem Let F /K be a constant field extension of F /K. Then the following hold: K is the full constant field of F. [F K(x)] = [F K (x)] for all x F K. F /F is unramified, i.e. no place of F ramifies in F. g F = g F. The conorm map Con F /F Cl(F ) Cl(F ) is injective. Corollary If F /K has finite class group, then h F divides h F. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 53 / 92

Genus 0 and 1 Function Fields

Genus 0 Function Fields We continue to assume that K is perfect. Theorem Let F /K be a function field of genus 0. Then the following hold: F /K is rational if and only if it has a rational (i.e. degree 1) place. If F /K is not rational, then F has a place of degree 2, and there exists x F with [F K(x)] = 2. Corollary For K algebraically closed, F /K is rational if and only if F has genus 0. Example Suppose K does not contain a square root of 1, and let F = K(x, y) where x 2 + y 2 = 1. Then F has genus 0 but is not rational. Remark Every genus 0 function field has class number 1. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 55 / 92

Genus 1 Function Fields Definition A function field F /K is elliptic if it has genus 1 and a rational place. Corollary For K algebraically closed, F /K is elliptic if and only if F has genus 1. Example Suppose K does not contain a square root of 1, and let F = K(x, y) where x 2 + y 4 = 1. Then F has genus 1 but is not elliptic. Theorem If F /K is elliptic, then there exist x, y F such that F = K(x, y) and y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6. for some a 1, a 2, a 3, a 4, a 6 K. This equation defines an elliptic curve in Weierstrass form. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 56 / 92

Short Weierstrass Form Consider y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6. If char(k) 2, then completing the square for y, i.e. substituting y by y (a 1 x + a 3 )/2 leaves F /K unchanged and produces an equation of the form y 2 = x 3 + b 2 x 2 + b 4 x + b 6 (b 2, b 4, b 6 K). If in addition char(k) 3, then completing the cube for x, i.e. substituting x by x b 2 /3 leaves F /K unchanged and produces an equation of the form y 2 = x 3 + Ax + B (A, B K). This is an elliptic curve in short Weierstrass form. Similarly, if char(k) = 2, one can always convert a (long) Weierstrass form to an equation of the form y 2 + y = cubic polynomial in x or y 2 + xy = cubic polynomial in x. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 57 / 92

Interlude Why Genus? Brief excursion into the topology and geometry of function fields. In topology, the genus g of a connected, orientable surface is the number of handles on it (or holes in it). It is the maximum number of (closed non-intersecting) cuts that are possible without disconnecting the surface. For example: A sphere has genus 0 A doughnut (torus) has genus 1, as does a coffee mug with a handle A pretzel (3-dimensional figure 8) has genus 2 Geometrically, over an algebraically closed field K, places of a function field F /K correspond one-to-one to the points on the unique non-singular plane curve defining F /K. A rational (i.e. genus 0) function field over C corresponds to the projective line P 1 (C), which is a sphere and thus a genus 0 object. An elliptic (i.e. genus 1) curve over C corresponds to the plane C 2 modulo its period lattice which is a torus and thus a genus 1 object. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 58 / 92

P 1 (F ) as an Abelian Group Theorem Let F /K be an elliptic function field, and fix a rational place P P 1 (F ). Then the injection Φ P 1 (F ) Cl 0 (F ) via P [P P ] is a bijection. Corollary Every degree zero divisor class of F /K has a unique representative of the form [P P ] with P P 1 (F ). The set P 1 (F ) becomes an abelian group (and Φ a group isomorphism) under the addition law P Q = R [P P ] + [Q P ] = [R P ]. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 59 / 92

Points on an Elliptic Curve Consider E y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6. Definition The set of (K-)rational points on E is E(K) = { (x 0, y 0 ) K K y0 2 + a 1x 0 y 0 + a 3 y 0 = x0 3 + a 2x0 2 + a 4x 0 + a 6 } { }. The point arises from the homogenization of E: E H y 2 z + a 1 xyz + a 3 yz 2 = x 3 + a 2 x 2 z + a 4 xz 2 + a 6 z 3. Points on E H : [x y z] [0 0 0] normalized to last non-zero entry = 1. Points on E Points on E H (x, y) [x y 1] (x/z, y/z) [x y z] when z 0 [0 1 0] Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 60 / 92

Point Arithmetic Theorem (Bezout) Two curves of respective degrees m and n intersect in exactly mn points (counting point multiplicities). Corollary A line intersects an elliptic curve in exactly 3 points. Group Law on E(K) (additive, abelian): Identity:. Inverses: p is defined as the third point of intersection of the vertical line through p and with E. Addition: If p q, then r is defined as the third point of intersection of the secant line through p and q with r. If p = q, then r is defined as the third point of intersection of the tangent line at p to E. Must then invert r to obtain r. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 61 / 92

An Elliptic Curve and a Point E y 2 = x 3 5x over Q, p = (-1, -2) E(Q) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 62 / 92

Inversion on E E y 2 = x 3 5x over Q, p = ( 1, 2) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 The vertical line through p and is x = 1 Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 63 / 92

Inversion on E E y 2 = x 3 5x over Q, p = ( 1, 2) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 It intersects E in the third point p = ( 1, 2) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 64 / 92

Addition of Distinct Points on E E y 2 = x 3 5x over Q, p = ( 1, 2), q = (0, 0) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 The line through p and q is y = 2x Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 65 / 92

Addition of Distinct Points on E E y 2 = x 3 5x over Q, p = ( 1, 2), q = (0, 0) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 It intersects E in the third point r = (5, 10) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 66 / 92

Addition of Distinct Points on E E y 2 = x 3 5x over Q, p = ( 1, 2), q = (0, 0) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 The sum r = p + q is the inverse of r, i.e. r = (5, 10) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 67 / 92

Doubling on E E y 2 = x 3 5x over Q, p = ( 1, 2) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 The line tangent to E at p is y = 19 26 x 33 26 Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 68 / 92

Doubling on E E y 2 = x 3 5x over Q, p = ( 1, 2) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 It intersects E in the third point r = ( 9 4, 3 8 ) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 69 / 92

Doubling on E E y 2 = x 3 5x over Q, p = ( 1, 2) 10 5 0-3 -2-1 0 1 2 3 4 5 6-5 -10 The sum r = p + p is the inverse of r, i.e. r = ( 9 4, 3 8 ) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 70 / 92

Rational Points and Rational Places Recall the addition law on P 1 (F ): P Q = R [P P ] + [Q P ] = [R P ] [P] + [Q] [R] = [P ] Recall the addition law on E(K): p + q r =. Theorem Let (x 0, y 0 ) E(K) { }. Then exists a unique P (x0,y 0 ) P 1 (F ) such that supp(div(x x 0 )) supp(div(y y 0 )) = {P (x0,y 0 ), P }. The map Ψ E(K) P 1 (K) via (x 0, y 0 ) P (x0,y 0 ) and P is a group isomorphism. So we have group isomorphisms (E(K), point addition) Ψ (P 1 (F ), ) Φ (Cl 0 (F ), divisor addition) Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 71 / 92

Hyperelliptic Function Fields

Hyperelliptic Function Fields Definition A function field F /K is hyperelliptic if it has genus at least 2 and there exists x F such that [F K(x)] = 2. Properties: F /K is hyperelliptic if and only if there exists D Div(F ) with deg(d) = 2 and l(d) 2. Every genus 2 function field is hyperelliptic. Description: Write F = K(x, y) with [F K(x)] = 2. Then F /K(x) has a minimal polynomial of the form y 2 + h(x)y = f (x) where h(x) and f (x) are polynomials (after we make everything integral) and h(x) = 0 if K has characteristic 2. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 73 / 92

Hyperelliptic Curves A hyperelliptic function field of genus g is of the form F = K(x, y) where with the following properties: f (x), h(x) K[x]; C is irreducible over K(x); C y 2 + h(x)y = f (x) C is non-singular (or smooth), i.e. there are no simultaneous solutions to C and its partial derivatives with respect to x and y. deg(f ) = 2g + 1 or 2g + 2; If K has characteristic 2, then h(x) = 0 ; If K has characteristic 2, then deg(h) g when deg(f ) = 2g + 1, and h(x) is monic of degree g + 1 when deg(f ) = 2g + 2; C is a hyperelliptic curve of genus g over K. Remark: The case g = 1 and deg(f ) odd also covers elliptic curves. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 74 / 92

Examples Every hyperelliptic curve over a field K of characteristic 2 has the form y 2 = f (x) with f (x) K[x] squarefree. y 2 = x 5 5x 3 + 4x 1 over Q, genus g = 2 10 5 0-3 -2-1 0 1 2 3-5 -10 Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 75 / 92

Classification by to Splitting at Infinity Case 1: deg(f ) = 2g + 1 (odd). Then P ramifies in F. Case 2: deg(f ) = 2g + 2 (even) and sgn(f ) is a square in K when q is odd; sgn(f ) is of the form s 2 + s for some s K when q is even. Then P splits in F. Case 3: deg(f ) = 2g + 2 (even) and sgn(f ) is a non-square in K when q is odd; sgn(f ) is not of the form s 2 + s with s K when q is even. Then P is inert in F. The representation of F /K(x) by C is referred to as ramified, split, and inert according to these three cases, or alternatively as imaginary, real, and unusual. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 76 / 92

Model Properties Ramified representations are the function field analogue of imaginary quadratic number fields. Split representations are the function field analogue of real quadratic number fields. Inert representations have no number field analogue. The variable transformation x 1/(x a) and y y/(x a) g+1, with f (a) 0, converts a ramified representation of F /K(x) into a split or inert representation of F /K(x) without changing the underlying rational function field K(x). The same variable transformation, with f (a) = 0, converts an inert or split representation of F /K(x) into a ramified representation of F (a)/k(a)(x); note that this may require an extension of the constant field. Inert models F /K(x) become split when considered over a quadratic extension over K. They don t exist over algebraically closed fields. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 77 / 92

Degree Zero Divisors Let D Div 0 (F ). Ramified model: Con F /K(x) (P ) = 2 with deg( ) = 1. D = D 0 deg(d 0 ) with supp(d 0 ). This gives rise to a group isomorphism from Div 0 (F ) onto the group of finite divisors {D Div(F ) supp(d)}. Split model: Con F /K(x) (P ) = + + with deg( + ) = deg( ) = 1. D = D 0 deg(d 0 ) n( + ) with +, supp(d 0 ) and n Z. This gives rise to a surjective group homomorphism from Div 0 (F ) onto the group of finite divisors {D Div(F ) +, supp(d)} with kernel +. Henceforth concentrate on finite divisors. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 78 / 92

Semi-Reduced and Reduced Divisors Definition A finite divisor D of F is semi-reduced if it is effective, i.e. v P (D) 0 for all P P(F ), and co-norm-free, i.e. can t be written as D = E + A where A is the con-norm of some divisor of K(x). D is reduced if it is semi-reduced and deg(d) g. Proposition A finite divisor is semi-reduced if and only if D is effective and for any P supp(d), the following hold: If P K is inert in F, then v P (D) = 0. If P K is ramified in F, then v P (D) = 1. If P K splits in F, say as P + Q, then either v P (D) = 0 or v Q (D) = 0. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 79 / 92

More on Semi-Reduced Divisors Lemma Every class in Cl 0 (F ) contains a divisor whose finite part is semi-reduced. Theorem Every class in Cl 0 (F ) contains a divisor D whose finite part D 0 is reduced. If [F K(x)] is ramified, then D is unique. If [F K(x)] is split, then there is a unique such divisor D with 0 n g deg(d 0 ). All but at most g classes have n = 0, i.e. D = D 0 deg(d 0 ) +. Remark If [F K(x)] is split, then the divisors D = D 0 deg(d 0 ) + form the infrastructure of F /K(x) (more on that later). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 80 / 92

Mumford Representation of Places Write F = K(x, y)] with y 2 + h(x)y = f (x). Lemma Let P be a finite place of F, and write P K(x) = P p(x) with p(x) K[x] monic and irreducible. Suppose P p(x) ramifies or splits in F. Then there exists a polynomial v(x) K[x], unique modulo p(x), with v(x) 2 + h(x)v(x) f (x) (mod p(x)) and v P (v + y) > 0.. Conversely, let p(x), v(x) K[x] with p(x) be monic and irreducible, and v(x) 2 + h(x)v(x) f (x) (mod p(x)). Then P p(x) ramifies or splits in F and there exists a unique finite place P of F lying above P p(x) with v P (v + y) > 0. Corollary The polynomials (p(x), v(x) mod p(x)) as above are in one-to-one correspondence with the finite places P of F such that P K(x) ramifies or splits in F. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 81 / 92

Mumford Representation of of Divisors Theorem Let D be a semi-reduced divisor of F.Then there exist u(x), v(x) K[x], with u(x) unique and v(x) unique modulo u(x), such that u(x) is monic; v(x) 2 + h(x)v(x) f (x) (mod u(x)); vp (u) > 0 and v P (y + v) > 0 for all P supp(d). Conversely, let u(x), v(x) K[x] with u(x) monic and v(x) 2 + h(x)v(x) f (x) (mod u(x)). Then the divisor D = n P P where n P = min{v P (u), v P (y + v)} for all finite places P of F is semi-reduced. Corollary and Definition The polynomials (u(x), v(x) mod u(x)) as above are in one-to-one correspondence with the semi-reduced divisors D of F. They are the Mumford representation of D, and we write D = (u, v). Note also that deg(d) = deg(u). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 82 / 92

Examples of Mumford Representations The Mumford representation of the trivial divisor is (1, 0). Suppose K has characteristic 2 and F = K(x, y) with y 2 = f (x) square-free. Let P P(F ) be ramified, and write P K(x) = P p(x). Then f (x) 0 (mod p(x)), and P has Mumford representation (p(x), 0). Every finite point (x 0, y 0 ) on a hyperelliptic curve corresponds to a rational place P of the function field of C with Mumford representation (x x 0, y 0 ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 83 / 92

Semi-reduced Divisors and K[x, y]-ideals Theorem Let u(x), v(x) K[x]. Then the following are equivalent: (u, v) is the Mumford representation of a reduced divisor of F. The K[x]-submodule of K[x, y] of rank 2 generated by u(x) and v(x) + y is an ideal of K[x, y]. v(x) 2 + h(x)v(x) f (x) (mod u(x)). Remark If F /K(x) is ramified, then the bijection between semi-reduced divisors and K[x, y]-ideals of the form above extends to a group isomorphism from Cl 0 (F ) onto the ideal class group of K[x, y]. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 84 / 92

Arithmetic in Cl 0 (F ) for Ramified Models F /K(x) Goal: efficient arithmetic on Cl 0 (F ) via unique reduced divisor class representatives in Mumford representations when F /K(x) is ramified: [D 1 ] + [D 2 ] = [D 1 D 2 ] where D 1 and D 2 are the unique reduced divisors in their respective classes; D 1 D 2 is the unique reduced divsor in the class of D 1 + D 2 ; All three divisors are given in Mumford representation. Road map: 1 First compute a semi-reduced divisor D equivalent to D 1 + D 2 in Mumford representation. 2 Then reduce D, i.e. compute the Mumford representation of the unique reduced divisor D 1 D 2 equivalent to D. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 85 / 92

Divisor Addition, Simplest Case Consider a ramified hyperelliptic function field F = K(x, y) where y 2 + h(x)y = f (x). Let D 1 = (u 1, v 1 ) and D 2 = (u 2, v 2 ). Proposition D 1 + D 2 is semi-reduced if and only if gcd(u 1, u 2, v 1 + v 2 + h) = 1. In this case, D 1 + D 2 = (u, v) where u = u 1 u 2, v 1 (mod u 1 ), v v 2 (mod u 2 ). Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 86 / 92

Divisor Addition, General Case Theorem Let D 1 = (u 1, v 1 ) and D 2 = (u 2, v 2 ) be semi-reduced divisors. Then D 1 + D 2 = D + div(s) where D = (u, v) is a semi-reduced divisor, and s, u, v K[x] are computed as follows: 1 Let s = gcd(u 1, u 2, v 1 + v 2 + h) = au 1 + bu 2 + c(v 1 + v 2 h) where a, b, c K[x] are computed using the extended euclidean algorithm. 2 Set u = u 1u 2 s 2. 3 Set v au 1v 2 + bu 2 v 1 + c(v 1 v 2 + f ) s (mod u). Note that even when D 1 and D 2 are reduced, D is generally not reduced! Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 87 / 92

Divisor Reduction Let D = (u, v) be a semi-reduced divisor with deg(u) > g. Set u = f + hv v 2, v h v (mod u ). u Properties: The divisor D = (u, v ) is equivalent to D. If deg(u) g + 2, then deg(u ) deg(u) 2. If deg(u) = g + 1, then deg(u ) g. Remarks If we repeatedly compute (u, v ) and substitute these values for u and v, we will obtain the reduced divisor D = (u, v ) equivalent to D. The number of these reduction steps required to obtain D is (deg(u) g)/2. If D was obtained by addition of two reduced divisors, then deg(u) 2g, and hence the number of reduction steps is g/2. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 88 / 92

Inert Models One infinite degree 2 place. Properties: If F /K(x) is inert and L is a quadratic extension of K, then FL/L(x) is split. Only the finite divisors of even degree correspond to degree zero divisors D 0 (deg(d 0 )/2). If K = F q, then very degree zero divisor class contains either a unique reduced representative or q + 1 almost reduced divisors (semi-reduced and deg(d 0 ) = g + 1, Artin 1924). Reduction finds the reduced or an almost reduced representative. For the latter case, Artin provided a procedure for finding the other q almost reduced equivalent divisors. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 89 / 92

Split Models Two infinite places +,, both of degree 1. Theorem (Paulus & Rück 1999) Every degree zero divisor class has a unique representative of the form D 0 deg(d 0 ) + n( + ), where D is reduced and 0 n g deg(d 0 ). Divisor class addition: Cantor plus adjustment steps (usually g/2 of them) to get n to the correct range. Infrastructure: Divisors with n = 0 ( almost a group) Theorem (Galbraith, Harrison & Mireles Morales 2008) Every degree zero divisor class has a unique balanced representative D 0 + deg(d 0 ) + n( + ), where D is reduced and g/2 n g/2 deg(d 0 ). Divisor class addition: Cantor with occasionally some balancing steps. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 90 / 92

Why Consider Split Representations? Advantages: A ramified representation need not always exist. Construction methods (e.g. for cryptography) don t always produce ramified models. Ramified F /K(x) models can be converted to split models F /K(x), but the reverse direction is only possible over a base field that contains a rational point. Mathematically interesting. Less researched than ramified models. Disadvantages: Mathematically more complicated than ramified models. Arithmetic is slightly slower. Less researched than ramified models. Renate Scheidler (Calgary) Number Theory in Function Fields UNGC, Summer 2016 91 / 92

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